Finite Group Databasebeta

← #34005254

Subgroups — #34005254

Order 18688.

630 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625787C211yesmaximal, characteristic
2#22625787C211yesmaximal, characteristic
3#22624809C214yesmaximal, complemented
4C2×#17988456C211yesmaximal, characteristic
5C2×#14556658C2212yes—
6C2×#14556658C2212yes—
7C2×C22⋊#10596833—21no—
8C2×C22⋊#10596833—21no—
9#17989453—21no—
10#17989453—21no—
11C22×C4⋊Dic73C411yescharacteristic
12C22×#11924007—21no—
13C22×#11924007—21no—
14C22×C4×Dic73C2211yescharacteristic
15#17988456—21no—
16#17988456—21no—
17#17988456C412yes—
18#17988456C2212yes—
19C22×C4⋊C4×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—24no—
23#14556658—24no—
24C22⋊#10596833—42no—
25C22⋊#10596833—42no—
26C23×Dic73—21no—
27C23×Dic73D411yescharacteristic
28C23×Dic73D411yescharacteristic
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73—42no—
31C2×C4⋊Dic73C2×C412yes—
32C2×#11924007—22no—
33C2×#11924007—22no—
34C2×#11924007—22no—
35C2×#11924007—22no—
36C2×#11924007—44no—
37C2×#11924007—42no—
38C2×#11924007—42no—
39C2×C4×Dic73—24no—
40C2×C4×Dic73C2×C411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73C2311yescharacteristic
44C23×C4×C73—21no—
45C23×C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—42no—
48C2×C4⋊C4×C73C2×C412yes—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C2×#10596833—42no—
55C22×Dic73—22no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—22no—
59C22×Dic73—22no—
60C22×Dic73—42no—
61C22×Dic73—42no—
62C22×Dic73—42no—
63C22×Dic73C22⋊C411yescharacteristic
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73D812yes—
66C22×Dic73SD1612yes—
67C22×Dic73SD1612yes—
68C22×Dic73Q1612yes—
69C22×Dic73C2×D411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C4⋊Dic73—42no—
72C4⋊Dic73—42no—
73#11924007—44no—
74#11924007—44no—
75#11924007—42no—
76#11924007—42no—
77#11924007—44no—
78C4×Dic73—22no—
79C4×Dic73—22no—
80C4×Dic73—22no—
81C4×Dic73#4114yes—
82C24×C73C22⋊C411yescharacteristic
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—42no—
88C22×C4×C73—42no—
89C22×C4×C73—42no—
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22×C411yescharacteristic
92C4⋊C4×C73—42no—
93C4⋊C4×C73—42no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73#10112yes—
99C23×C73#10212yes—
100C23×C73#10312yes—
101C23×C73C2×C22⋊C411yescharacteristic
102C2×C4×C73—28no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—84no—
111C2×C4×C73#10012yes—
112C2×C4×C73#11712yes—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—24no—
118C2×Dic73—24no—
119C2×Dic73—24no—
120C2×Dic73—24no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—84no—
125C2×Dic73#10212yes—
126C2×Dic73#10312yes—
127C2×Dic73#12312yes—
128C2×Dic73#12312yes—
129C2×Dic73C2×D811yescharacteristic
130C2×Dic73C2×SD1611yescharacteristic
131C2×Dic73C2×SD1611yescharacteristic
132C2×Dic73C2×Q1611yescharacteristic
133C2×Dic73AGL(1,ℤ/8)12yes—
134C2×Dic73#13712yes—
135#10596833—82no—
136#10596833—82no—
137C22×C73—22no—
138C22×C73—22no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—22no—
142C22×C73—22no—
143C22×C73#33214yes—
144C22×C73#38111yescharacteristic
145C22×C73C2×#10011yescharacteristic
146C22×C73C2×#10111yescharacteristic
147C22×C73C2×#10211yescharacteristic
148C22×C73C2×#10311yescharacteristic
149C22×C73#41811yescharacteristic
150C22×C73#41911yescharacteristic
151C4×C73—22no—
152C4×C73—22no—
153C4×C73—82no—
154Dic73—22no—
155Dic73—22no—
156Dic73—82no—
157Dic73#48012yes—
158Dic73#48112yes—
159Dic73#48312yes—
160Dic73#48412yes—
161#8724—731nomaximal, Sylow
162C2×C73—24no—
163C2×C73C2×#33211yescharacteristic
164C2×C73#154511yescharacteristic
165C2×C73#154811yescharacteristic
166C2×C73#155611yescharacteristic
167C2×C73#182111yescharacteristic
168C2×C73#182311yescharacteristic
169C2×C73#182411yescharacteristic
170C2×#384—731no—
171#1797—731no—
172#1797—731no—
173#1296—734no—
174C73#872411yescharacteristic, complemented, Sylow
175C22×C4⋊C4—731no—
176C22×C4⋊C4—1461no—
177C22×C4⋊C4—1461no—
178C22×C4⋊C4#6440811yescharacteristic
179C22×C42—731no—
180C2×C4⋊C8—732no—
181C2×C4⋊C8—732no—
182C2×C22⋊C8—1461no—
183C2×C22⋊C8—1461no—
184#384—732no—
185#384—732no—
186#384—1461no—
187#384—1461no—
188#336—1461no—
189#336—1461no—
190C23×C4—21no—
191C23×C4—731no—
192C23×C4—731no—
193C23×C4—1461no—
194C23×C4C2×#6440811yescharacteristic
195C22×C8—1462no—
196C22×C8—1462no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—2922no—
200C2×C4⋊C4—2922no—
201C2×C4⋊C4—2924no—
202C2×C4⋊C4—2922no—
203C2×C4⋊C4—732no—
204C2×C4⋊C4—1462no—
205C2×C4⋊C4—1462no—
206C2×C4⋊C4—1462no—
207C2×C4⋊C4—1462no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4C2×#6440812yes—
210C2×C42—731no—
211C2×C42—731no—
212C2×C42—731no—
213C2×C42—731no—
214C2×C42—1464no—
215C4⋊C8—1464no—
216C4⋊C8—1464no—
217C22⋊C8—2922no—
218C22⋊C8—2922no—
219C24C22⋊#6440811yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—2922no—
228C22×C4—2922no—
229C22×C4—2922no—
230C22×C4—731no—
231C22×C4—731no—
232C22×C4—732no—
233C22×C4—732no—
234C22×C4—732no—
235C22×C4—732no—
236C22×C4—731no—
237C22×C4—731no—
238C22×C4—1462no—
239C22×C4—1462no—
240C22×C4—1462no—
241C22×C4—1462no—
242C22×C4—1462no—
243C22×C4C22⋊#6440811yescharacteristic
244C22×C4C22×#6440811yescharacteristic
245C2×C8—2922no—
246C2×C8—2922no—
247C2×C8—2922no—
248C2×C8—2922no—
249C2×C8—2922no—
250C2×C8—2922no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—2922no—
254C4⋊C4—2922no—
255C4⋊C4—2922no—
256C4⋊C4—2924no—
257C4⋊C4—2924no—
258C4⋊C4—2924no—
259C4⋊C4—2922no—
260C42—734no—
261C42—1462no—
262C42—1462no—
263C42—1462no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1455665212yes—
269C23#1455665312yes—
270C23#1455665412yes—
271C23C2×C22⋊#6440811yescharacteristic
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—84no—
281C2×C4—2922no—
282C2×C4—2922no—
283C2×C4—2922no—
284C2×C4—5844no—
285C2×C4—732no—
286C2×C4—732no—
287C2×C4—731no—
288C2×C4—731no—
289C2×C4—731no—
290C2×C4—731no—
291C2×C4—732no—
292C2×C4—732no—
293C2×C4—732no—
294C2×C4—732no—
295C2×C4—1464no—
296C2×C4—1464no—
297C2×C4—1464no—
298C2×C4—1464no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1461no—
303C2×C4#1455665112yes—
304C2×C4#1455667012yes—
305C8—5842no—
306C8—5842no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22#1798944614yes—
314C22#1798953011yescharacteristic
315C22C2×#1455665111yescharacteristic
316C22C2×#1455665211yescharacteristic
317C22C2×#1455665311yescharacteristic
318C22C2×#1455665411yescharacteristic
319C22#1798957911yescharacteristic
320C22#1798958011yescharacteristic
321C4—22no—
322C4—22no—
323C4—82no—
324C4—5842no—
325C4—732no—
326C4—732no—
327C4—732no—
328C4—732no—
329C4—1462no—
330C4—1462no—
331C2—24no—
332C2C2×#1798944611yescharacteristic
333C2#2262543811yescharacteristic
334C2#2262544411yescharacteristic
335C2#2262546011yescharacteristic
336C2#2262583011yescharacteristic
337C2#2262583411yescharacteristic
338C2#2262583511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8724
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22624809C2semidirect
174. C73#8724semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226257872
#226248092
C2×#179884562
#872473